Slide 1 / 162
Slide 2 / 162 Algebra II Trigonometric Functions 2015-12-17 www.njctl.org
Slide 3 / 162 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc Length & Area of a Sector Unit Circle Graphing Trigonometric Identities
Slide 4 / 162 Radians & Degrees and Co-Terminal Angles Return to Table of Contents
Slide 5 / 162 A few definitions: A central angle of a circle is an angle whose vertex is the center of the circle. An intercepted arc is the part of the circle that includes the points of intersection with the central angle and all the points in the interior of the angle. central angle intercepted arc
Slide 6 / 162 Radians and Degrees One radian is the measure of a central angle that intercepts an arc whose length is equal to the radius of the circle. There are, or a little more than 6, radians in a circle. Click on the circle for an animated view of radians.
Slide 7 / 162 Converting from Degrees to Radians There are 360 in a circle. Therefore 360 = 2 radians 2 1 = 360 = 180 radians Use this conversion factor to covert degrees to radians. Example: Convert 50 and 90 to radians. 50 = 5 radians 180 18 90 = radians 180 2
Slide 8 / 162 Converting from Radians to Degrees 2 radians = 360 360 1 radian = = 180 2 degrees Use this conversion factor to covert radians to degrees. Example: Convert and to radians 4 4 180 = 45 180 = 180
Slide 9 / 162 Converting between Radians and Degrees Convert degrees to radians
Slide 9 (Answer) / 162 Converting between Radians and Degrees Convert degrees to radians Answer [This object is a pull tab]
Slide 10 / 162 Converting between Radians and Degrees Convert radians to degrees radians radians radians
Slide 10 (Answer) / 162 Converting between Radians and Degrees Convert radians to degrees radians radians Answer radians [This object is a pull tab]
Slide 11 / 162
Slide 11 (Answer) / 162
Slide 12 / 162
Slide 12 (Answer) / 162
Slide 13 / 162
Slide 13 (Answer) / 162
Slide 14 / 162 4 Convert radians to degrees:
Slide 14 (Answer) / 162 4 Convert radians to degrees: Answer 67.5 0 [This object is a pull tab]
Slide 15 / 162 Angles Terminal side Initial side Terminal side Initial side Angle Angle in standard position An angle is formed by rotating a ray about its endpoint. The starting position is the initial side and the ending position is the terminal side. When, on the coordinate plane, the vertex of the angle is the origin and the initial side is the positive x-axis, the angle is in standard position.
Slide 16 / 162 Positive Angle - terminal side rotates in a counterclockwise direction Negative Angle - terminal side rotates in a clockwise direction α = - 37
Slide 17 / 162 Drawing angles in standard position 310 500 40 Each quadrant is 90, and 310 is 40 more than 270, so the terminal side is 40 past the negative y-axis. 500 is 140 more than 360, so the angle makes a complete revolution counterclockwise and then another 140.
Slide 18 / 162 Coterminal Angles Angles that have the same terminating side are coterminal. To find coterminal angles add or subtract multiples of 360 for degrees and 2 for radians. Example: Find one positive and one negative angle that are terminal with 75. 75 + 360 = 435 75-360 = -285-285 435 75
Slide 19 / 162 5 Which angles are coterminal with 40? (Select all that are correct.) A 320 B -320 C 400 D -400
Slide 19 (Answer) / 162 5 Which angles are coterminal with 40? (Select all that are correct.) A 320 B -320 C 400 D -400 Answer B, C [This object is a pull tab]
Slide 20 / 162 6 Which graph represents 425? A B C D
Slide 20 (Answer) / 162 6 Which graph represents 425? A B Answer A C D [This object is a pull tab]
Slide 21 / 162 7 Which graph represents? A B C D
Slide 21 (Answer) / 162 7 Which graph represents? A B Answer D C D [This object is a pull tab]
Slide 22 / 162 8 Which angle is NOT coterminal with -55? A 305 B 665 C -415 D -305
Slide 22 (Answer) / 162 8 Which angle is NOT coterminal with -55? A 305 B 665 C -415 D -305 Answer A, B, C [This object is a pull tab]
Slide 23 / 162 9 Which angle is coterminal with? A B C D
Slide 23 (Answer) / 162 9 Which angle is coterminal with? A B C Answer B [This object is a pull tab] D
Slide 24 / 162 Arc Length & Area of a Sector Return to Table of Contents
Slide 25 / 162 Arc length and the area of a sector (Measured in radians) r arc length s sector Arc length: s = r Area of sector: A = How do these formulas relate to the area and the circumference of a circle?
Slide 26 / 162 Who is getting more pie? Who is getting more of the crust at the outer edge? 40 45 Emily's slice is cut from a 9 inch pie. Chester's slice is cut from an 8 inch pie. (Assume both pies are the same height.) (Try to work this out in your groups. The solution is on the next slide)
Slide 27 / 162 40 45 click The top of Emily's piece has an area of click The top of Chester's piece has an area of Emily's crust has a length of Chester's crust has a length of
Slide 28 / 162 10 What is the top surface area of this slice of pizza from an 18-inch pie? 45
Slide 28 (Answer) / 162 10 What is the top surface area of this slice of pizza from an 18-inch pie? 45 Answer 1 2 92 = 81 31.8 in 4 8 2 [This object is a pull tab]
Slide 29 / 162 11 What is the arc length of the outer edge of this slice of pizza from an 18-inch pie? 45
Slide 29 (Answer) / 162 11 What is the arc length of the outer edge of this slice of pizza from an 18-inch pie? 45 Answer 9 4 7.1 in [This object is a pull tab]
Slide 30 / 162 12 If the radius of this circular saw blade is 10 inches and there are 40 teeth on the blade, how far apart are the tips of the teeth?
Slide 30 (Answer) / 162 12 If the radius of this circular saw blade is 10 inches and there are 40 teeth on the blade, how far apart are the tips of the teeth? Answer 2 40 10 1.6 inches [This object is a pull tab]
Slide 31 / 162 13 Challenge Question: Given a dart board as shown. If a dart thrown randomly lands somewhere on the board, what is the probability that it will land on a red region? 4 in 8 inches
Slide 31 (Answer) / 162 13 Challenge Question: Given a dart board as shown. If a dart thrown randomly lands somewhere on the board, what is the probability that it will land on a red region? each of the inner red regions: each of the outer red regions: 4 in rea of all red regions: 8 inches the entire dart board: ility: [This object is a pull tab] or about 10%
Slide 32 / 162 Unit Circle Return to Table of Contents
Slide 33 / 162 The circle x 2 + y 2 = 1, with center (0,0) and radius 1, is called the unit circle. The Unit Circle Quadrant II: x is negative and y is positive (0,1) 1 Quadrant I: x and y are both positive (-1,0) Quadrant III: x and y are both negative (0,-1) (1,0) Quadrant IV: x is positive and y is negative
θ a (a,b) b Slide 34 / 162 The unit circle allows us to extend trigonometry beyond angles of triangles to angles of all measures. (-1,0) (0,1) 1 (1,0) In this triangle, b sin#= 1 = b cos# = a 1 = a so the coordinates of (a,b) are also (0,-1) (cos#, sin#) For any angle in standard position, the point where the terminal side of the angle intercepts the circle is called the terminal point.
Slide 35 / 162 In this example, the terminal point is in Quadrant IV. 0.57-55 1 0.82 If we look at the triangle, we can see that sin(-55 ) = 0.82 cos(-55 ) = 0.57 EXCEPT that we have to take the direction into account, and so sin(-55 ) is negative because the y value is below the x-axis. For any angle θ in standard position, the terminal point has coordinates (cosθ, sinθ).
Slide 36 / 162 Click the star below to go to the Khan Academy Unit Circle Manipulative try some problems:
Slide 37 / 162 What are the coordinates of point C? In this example, we know the angle. Using a calculator, we find that cos 44.72 and sin 44.69, so the coordinates of C are approximately (0.72, 0.69). 1 Note that 0.72 2 + 0.69 2 1!
Slide 38 / 162 The Tangent Function Recall SOH-CAH-TOA sin = # opp hyp cos # = adj hyp tan # = opp adj opposite side hypotenuse # adjacent side It is also true that tan = sin # # cos. # Why? opp hyp adj hyp opp hyp hyp adj = = opp = tan adj #
Slide 39 / 162 Angles in the Unit Circle Example: Given a terminal point on the unit circle (- ). Find the value of cos, sin and tan of the angle. Solution: Let the angle be. x = cos, so cos =. y = sin, so sin =. tan = = = = (Shortcut: Just cross out the 41's in the complex fraction.)
Slide 40 / 162 Example: Given a terminal point csc#. Note the "hidden" Pythagorean Triple, 8, 15, 17)., find #, tan# and To find #, use sin -1 or cos -1 : sin -1 ( ) = # # # 28.1 tan# = sin#/ cos# tan # = csc# = 1/ sin# csc # =
Slide 41 / 162 Example: Find the x-value of point A, θ and the tan θ. For every point on the circle, θ A 22.3-22.3 5 (, - 13 ) Since x is in quadrant III, x = - 12 5 sin -1 (- 13 ) -22.3, BUT θ is in quadrant III, so θ = 180 + 22.3 = 202.3 (notice how 202.3 and -22.3 have the same sine) sin θ tan θ = cos θ = = 5 12 13
Slide 42 / 162 Example: Given the terminal point of ( -5 / 13, -12 / 13 ). Find sin x, cos x, and tan x.
Slide 42 (Answer) / 162 Example: Given the terminal point of ( -5 / 13, -12 / 13 ). Find sin x, cos x, and tan x. Answer Solution: 1)Graph the point 2)Draw a triangle with the radius as the hypotenuse and one leg on the x-axis. 3) write your trig ratios cos x = -5 / 13 sin x = -12 / 13 tan x = 12 / 13-5 -12 13 ( -5, -12) [This object is a pull tab]
Slide 43 / 162 14 What is tan θ? 3 (- 5, ) A θ B C D
Slide 43 (Answer) / 162 14 What is tan θ? 3 (- 5, ) A θ B C D Answer D [This object is a pull tab]
Slide 44 / 162 15 What is sin θ? 3 (- 5, ) A θ B C D
Slide 44 (Answer) / 162 15 What is sin θ? 3 (- 5, ) A θ B C Answer B D [This object is a pull tab]
Slide 45 / 162 16 What is θ (give your answer to the nearest degree)? 3 (- 5, ) θ
Slide 45 (Answer) / 162 16 What is θ (give your answer to the nearest degree)? 3 (- 5, ) Answer θ cos-1(-.6) 127 [This object is a pull tab]
Slide 46 / 162 17 Given the terminal point, find tan x.
Slide 46 (Answer) / 162 17 Given the terminal point, find tan x. Answer [This object is a pull tab]
Slide 47 / 162
Slide 47 (Answer) / 162
Slide 48 / 162 Equilateral and isosceles triangles occur frequently in geometry and trigonometry. The angles in these triangles are multiples of 30 and 45. A calculator will give approximate values for the trig functions of these angles, but we often need to know the exact values. Isosceles Right Triangle Equilateral Triangle (the altitude divides the triangle into two 30-60-90 triangles)
Slide 49 / 162 Special Right Triangles (see Triangle Trig Review unit for more detail on this topic)
Slide 50 / 162 Special Triangles and the Unit Circle (-, ) (, ) 1 1-45 45 Multiples of 45 angles have sin and cos of ±, depending on the quadrant.
Slide 51 / 162 30 o 45 o 60 o 60 o 45 o 30 o 30 o 30 o 45 o 60 o 60 o 45 o
Slide 52 / 162 Drag the degree and radian angle measures to the angles of the circle: # 5# # 3# 7# 3# 0 # 2# 4 4 2 2 4 4 0 45 90 135 180 225 270 315 360
Slide 53 / 162 Fill in the coordinates of x and y for each point on the unit circle: (, ) (, ) 3# 4 # 2 # 4 (, ) (, ) # 2# 0 (, ) 0 1 (, ) 5# 4 3# 2 7# 4 (, ) -1 (, )
Slide 54 / 162 Special Triangles and the Unit Circle (, ) 1 (, ) 1 30 60 Angles that are multiples of 30 have sin and cos of ± and ±.
Slide 55 / 162 Drag the degree and radian angle measures to the angles of the circle: 0 30 60 90 120 150 180 210 240 270 300 330 360 5# 6 # 2 # 7# # 0 2# 4# # 3# 2# 11# 5# 6 3 3 2 3 6 6 3
Slide 56 / 162 Drag in the coordinates of x and y for each point on the unit circle: (, ) (, ) (, ) 2# 3 # 5# 6 # 2 (, ) # 3 (, ) # 6 2# 0 (, ) (, ) (, ) 7# 6 4# 3 (, ) 3# 2 5# 3 (, ) 11# 6 (, ) (, ) 0 1-1
Special Angles in Degrees Slide 57 / 162
Slide 58 / 162 Radian Values of Special Angles
Exact Values of Special Angles Slide 59 / 162
Put it all together... Slide 60 / 162
Slide 61 / 162 Exact values of special angles Complete the table below: Degrees 0 30 45 60 90 Radians sin θ cos θ tan θ
Slide 62 / 162
Slide 63 / 162
Slide 64 / 162 If we know one trig function value and the quadrant in which the angle lies, we can find the angle and the other trig values.
Slide 65 / 162 Example: If tan =, and sin < 0, find sin, cos and the value of. Solution: Since tan is positive and sin is negative, the terminal side of must be in Quadrant III. Draw a right triangle in Quadrant III. Use the Pythagorean Theorem to find the length of the hypotenuse: opp -3 adj hyp (Continued on next slide)
Slide 66 / 162 Once we know the lengths for each side, we can calculate the sin, cos and the angle. Used the signed numbers to get the correct values. sin = = opp -3 adj hyp cos = = Use any inverse trig function to find the angle. tan-1( ) 36.7. Because the angle is in QIII, we need to add 180 + 36.7 = 216.7, so 217.
Slide 67 / 162
Slide 67 (Answer) / 162
Slide 68 / 162
Slide 68 (Answer) / 162
Slide 69 / 162
Slide 69 (Answer) / 162
Slide 70 / 162 22 Which functions are positive in the second quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x
Slide 70 (Answer) / 162 22 Which functions are positive in the second quadrant? Choose all that apply. A B C D cos x sin x tan x sec x Answer B E E F csc x cot x [This object is a pull tab]
Slide 71 / 162 23 Which functions are positive in the fourth quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x
Slide 71 (Answer) / 162 23 Which functions are positive in the fourth quadrant? Choose all that apply. A cos x B C D sin x tan x sec x Answer A D E csc x F cot x [This object is a pull tab]
Slide 72 / 162 24 Which functions are positive in the third quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x
Slide 72 (Answer) / 162 24 Which functions are positive in the third quadrant? Choose all that apply. A B C cos x sin x tan x Answer C F D sec x E csc x [This object is a pull tab] F cot x
Slide 73 / 162 Graphing Trig Functions Return to Table of Contents
Slide 74 / 162 If you have Geogebra available on your computer, click the star below to download a geogebratube animated graph of the trig functions: (Once the webpage opens, click on Download)
Slide 75 / 162 Graphing the Sine Function, y = sin x Graph by creating a table of values of key points. One option is to use the set of values for x that are multiples of, and the corresponding values of y or sin x. (Remember, is just a bit more than 3.) Since the values are based on a circle, values will repeat.
Slide 76 / 162 Notice that the graph of y = sin x increases from 0 to 1, then decreases back to 0 and then to -1 and then goes back up to 0, as x increases from 0 to 2.
Slide 77 / 162 Graphing the Cosine Curve Make a table of values just as we did for sin. We could use any interval, but are choosing from 0 to 2. Since the values are based on a circle, values will repeat.
Slide 78 / 162 Notice that the graph of y = cos x starts at 1, decreases to -1 and then goes back up to 1 as x increases from 0 to 2.
Slide 79 / 162 Compare the graphs: y = sin x y = cos x How are they similar and how are they different?
Slide 80 / 162 Characteristics of y = sin x and y = cos x range: -1 y 1 amplitude = 1 period = 2 Domain: set of real numbers (x can be anything) Range: -1 y 1 Amplitude: one-half the distance from the minimum of the graph to the maximum or 1. The functions are periodic - the pattern repeats every 2 units.
Slide 81 / 162 Predict, Explore, Confirm 1. Using your prior knowledge of transforming functions, predict what happens to the following functions: 2. Using your graphing calculator, insert the parent function into and the transformed function into. Compare the graphs. 3. Do your conclusions match your predictions?
Slide 81 (Answer) / 162 Predict, Explore, Confirm Students have experience with transforming functions; however, 1. Using your prior knowledge they may of transforming need a reminder functions, of what predict what happens to the following functions: values cause the function to vertically stretch and shrink. Teacher Notes Encourage students to make predictions on their own first. As they explore they can begin to 2. Using your graphing calculator, insert the parent function into and discuss with classmates and the transformed function into. Compare the graphs. confirm [This object their is a pull results. tab] 3. Do your conclusions match your predictions?
Slide 82 / 162 y = a sin x or y = a cos x Amplitude is a positive number that represents one-half the difference between the minimum and the maximum values, or the distance from the midline to the maximum.
Slide 82 (Answer) / 162 y = a sin x or y = a cos x Teacher Notes Students should now recall that in the study of transforming parent functions, we learned "a" was a vertical stretch or shrink. They will now learn that for trig functions it is called the amplitude. Amplitude is a positive number that represents one-half the difference between the minimum and the maximum values, [This object or is the a pull tab] distance from the midline to the maximum.
Slide 83 / 162 Consider the graphs of y = sin x What do you notice about these y = 2sin x graphs? What does the value of y = sin x "a" do to the graph? y = sin x y = 2sin x y = sin x Name the amplitude of each graph.
Slide 83 (Answer) / 162 Consider the graphs of y = sin x What do you notice about these y = 2sin x graphs? What does the value of y = sin x "a" do to the graph? y = sin x y = Answer sin x y = 2sin x y = 2 sin x: 2 y = sin x: 1 y =.5 sin x:.5 [This object is a pull tab] Name the amplitude of each graph.
Slide 84 / 162 As shown in the graph below, the graph of y = -3cos xis a reflection over the x-axis of the graph of y = 3cos x. What is the amplitude of each function? y = 3cos x y = -3cos x The domain of each function is the set of real numbers and the range is {x -3 x 3}.
Slide 85 / 162 Sketch each graph on the interval from 0 to 2 : y = 4cos x y = -.25 sin x
Slide 86 / 162 25 What is the amplitude of y = 3cos x?
Slide 86 (Answer) / 162
Slide 87 / 162 26 What is the amplitude of y = 0.25cos x?
Slide 87 (Answer) / 162 26 What is the amplitude of y = 0.25cos x? Answer [This object is a pull tab]
Slide 88 / 162 27 What is the amplitude of y = -sin x?
Slide 88 (Answer) / 162
Slide 89 / 162 28 What is the range of the function y = 2sin x? A All real numbers B -2 < x < 2 C 0 x 2 D -2 x 2
Slide 89 (Answer) / 162 28 What is the range of the function y = 2sin x? A All real numbers B -2 < x < 2 C 0 x 2 D -2 x 2 Answer D [This object is a pull tab]
Slide 90 / 162 29 What is the domain of y = -3cos x? A All real numbers B -3 < x < 3 C 0 x 3 D -3 x 3
Slide 90 (Answer) / 162 29 What is the domain of y = -3cos x? A All real numbers B -3 < x < 3 C 0 x 3 D -3 x 3 Answer A [This object is a pull tab]
Slide 91 / 162 30 Which graph represents the function y = -2sin x? A B C D
Slide 91 (Answer) / 162 30 Which graph represents the function y = -2sin x? A B C D Answer B [This object is a pull tab]
Slide 92 / 162 31 What is the amplitude of the graph below?
Slide 93 / 162 Predict, Explore, Confirm 1. Using your prior knowledge of transforming functions, predict what happens to the following functions: 2. Using your graphing calculator, insert the parent function into and the transformed function into. Compare the graphs. 3. Do your conclusions match your predictions?
Slide 93 (Answer) / 162 Predict, Explore, Confirm Students have experience with transforming functions; however, 1. Using your prior knowledge of transforming functions, predict what they may need a reminder of what happens to the following functions: values cause the function to horizontally stretch and shrink. Teacher Notes Encourage students to make predictions on their own first. As they explore they can begin to 2. Using your graphing calculator, discuss insert with the classmates parent function and into and the transformed function into. confirm Compare the graphs. [This object their is a pull results. tab] 3. Do your conclusions match your predictions?
Slide 94 / 162 A periodic function is one that repeats its values at regular intervals. One complete repetition of the pattern is called a cycle. The period is the length of one complete cycle. The trig functions are periodic functions. The basic sine and cosine curves have a period of 2, meaning that the graph completes one complete cycle in 2 units.
Slide 95 / 162 y = sin bx or y = cos bx Consider the graphs of y = cos xand y = cos 2x. y = cos x one cycle y = cos 2x Notice that the graph of y = cos 2xcompletes one cycle twice as fast, or in units.
Slide 96 / 162 y = cos x completes 1 cycle in 2#. So the period is 2π. y = cos 2x completes 2 cycles in 2# or 1 cycle in #. The period is #. y = cos 0.5x completes a cycle in 4#. The period is 4#.
Slide 97 / 162 The period for y = cos bx or y = sin bx is P = 2 b y = cos x b = 1 2 P = 1 = 2 y = cos 2x b = 2 P = 2 2 = y = cos 0.5x b = 0.5 P = 2 0.5 = 4
Slide 98 / 162 32 What is the period of A B C D
Slide 98 (Answer) / 162 32 What is the period of A B C D Answer B [This object is a pull tab]
Slide 99 / 162 33 What is the period of A B C D
Slide 99 (Answer) / 162 33 What is the period of A B C D Answer D [This object is a pull tab]
Slide 100 / 162 34 What is the period of A B C D
Slide 100 (Answer) / 162 34 What is the period of A B C D Answer A [This object is a pull tab]
Slide 101 / 162 Sketch the graph of each function from x = 0 to x = 2. y = 2cos 3x y = cos x y = sin 2x y = -2cos 2x
Slide 101 (Answer) / 162 Sketch the graph of each function from x = 0 to x = 2. y = 2cos 3x Hint Hint: Once you have identified the cycle, divide one cycle in half and then each half in half. Use these divisions to graph the minimums, maximums and zeros of the graph. y = cos x [This object is a pull tab] y = sin 2x y = -2cos 2x
Slide 102 / 162 35 What is the period of the graph below? A B 2 C 3 2 D 2
Slide 102 (Answer) / 162 35 What is the period of the graph below? A B 2 C 3 2 Answer A D 2 [This object is a pull tab]
Slide 103 / 162 36 What is the period of the graph shown? A B C D 2 2 3 3
Slide 103 (Answer) / 162 36 What is the period of the graph shown? A B 2 C 2 3 Answer D [This object is a pull tab] D 3
Slide 104 / 162 37 What is the equation of this function? A B C D y = sin 3x y = cos 3x y = 3cos x y = 3sin x
Slide 104 (Answer) / 162 37 What is the equation of this function? A y = sin 3x Answer B B y = cos 3x C D y = 3cos x y = 3sin x [This object is a pull tab]
Slide 105 / 162 Predict, Explore, Confirm 1. Using your prior knowledge of transforming functions, predict what happens to the following functions: 2. Using your graphing calculator, insert the parent function into and the transformed function into. Compare the graphs. 3. Do your conclusions match your predictions?
Slide 105 (Answer) / 162 Predict, Explore, Confirm Students have experience with transforming functions; however, 1. Using your prior knowledge they may of transforming need a reminder functions, of what predict what happens to the following functions: values cause the function to shift horizontally and vertically. Teacher Notes Encourage students to make predictions on their own first. As they explore they can begin to 2. Using your graphing calculator, discuss insert with the classmates parent function and into and the transformed function into. Compare confirm [This object their is a pull results. graphs. tab] 3. Do your conclusions match your predictions?
Slide 106 / 162 Translating Sine and Cosine Functions Trig functions can be translated in the same way as any other function. The horizontal shift is called a phase shift. What are your conclusions from the graphing calculator activity?
Slide 106 (Answer) / 162 Translating Sine and Cosine Functions Answer Trig functions can be translated in the same way as any f(x other - h) moves function. the graph of f(x), the parent function, h units horizontally. The When horizontal h > 0, the shift function is called shifts a phase left and shift. when h < 0, the function shifts right. What are your conclusions from the graphing calculator activity? f(x) + k moves the graph of f(x), the parent function, k units vertically. When k > 0, the function shifts up and when k < 0, the function shifts down. [This object is a pull tab]
Horizontal or phase shift y = cos x Slide 107 / 162 Drag each equation to the matching graph y = cos (x + ) 2 Vertical shift y = sin x y = sin x + 2 k
Slide 108 / 162
Slide 108 (Answer) / 162
Slide 109 / 162
Slide 109 (Answer) / 162
Slide 110 / 162 Consider the graphs of and (which is which?) In order to determine the phase shift, the coefficient of x must be factored out. In shift is. In, the 2 is factored out. The phase, when the 2 is factored out, we get. The phase shift is.
Slide 111 / 162 Another way to determine the phase shift is to set the quantity inside the parenthesis equal to 0 and solve for the variable. Example: Set Solve for x: So, the phase shift is 2.
Slide 112 / 162
Slide 112 (Answer) / 162
Slide 113 / 162
Slide 113 (Answer) / 162
Slide 114 / 162 Vertical Shift y= sin (x) + k or y= cos (x) + k The k moves the graph up or down. The graph below is of the equation y = 2 sin (3x). The midline of this graph is the horizontal line y = 0. Sketch the graph of y = 2 sin (3x) + 1.
Slide 115 / 162 42 What is the vertical shift in
Slide 115 (Answer) / 162 42 What is the vertical shift in Answer (down 5 units) [This object is a pull tab]
Slide 116 / 162 43 What is the vertical shift in
Slide 116 (Answer) / 162 43 What is the vertical shift in Answer (up 7 units) [This object is a pull tab]
Slide 117 / 162 44 What is the vertical shift in
Slide 117 (Answer) / 162 44 What is the vertical shift in Answer (up 1 unit) [This object is a pull tab]
Slide 118 / 162
Slide 119 / 162 Graphing a Sine or Cosine Function: Step 1: Identify the amplitude, period, phase shift and vertical shift. Step 2: Draw the midline (y = k) Step 3: Find 5 key points - maximums, minimums and points on the midline Step 4: Draw the graph through the 5 points.
Slide 120 / 162 Example: Step 1: Amplitude: -1 = 1 Period: Phase Shift: Vertical Shift: 2 (up 2)
Slide 121 / 162 Step 2: Draw the midline y = 2 Step 3: Find the 5 key points Note: for x, adding the cycle, 3 by 4. comes from dividing For y, adding and subtracting 1 comes from the amplitude.
Step 4: Graph Slide 122 / 162
You try: Slide 123 / 162
Slide 123 (Answer) / 162 You try: Answer [This object is a pull tab]
Slide 124 / 162
Slide 124 (Answer) / 162
Slide 125 / 162
Slide 126 / 162
Slide 126 (Answer) / 162
Slide 127 / 162
Slide 127 (Answer) / 162
Slide 128 / 162 49 What is the amplitude of this cosine graph?
Slide 128 (Answer) / 162
Slide 129 / 162 50 What is the period of this cosine graph? (use 3.14 for pi)
Slide 129 (Answer) / 162 50 What is the period of this cosine graph? (use 3.14 for pi) Answer [This object is a pull tab]
Slide 130 / 162 51 What is the vertical shift of this cosine graph?
Slide 130 (Answer) / 162 51 What is the vertical shift of this cosine graph? Answer [This object is a pull tab]
Slide 131 / 162 52 Which of the following of the following is an equation for the graph? A B C D
Slide 131 (Answer) / 162
Slide 132 / 162 The equation y = 4.2cos (π/6(x - 1)) + 13.7 can be used to model the average temperature of Wellington, NZ in degrees Celsius, where x represents the month, 1-12. Sketch the graph of this equation. What is the average temperature in June?
Slide 132 (Answer) / 162 The equation y = 4.2cos (π/6(x - 1)) + 13.7 can be used to model the average temperature of Wellington, NZ in degrees Celsius, where x represents the month, 1-12. Sketch the graph of this equation. What is the average temperature in June? Answer The avg temp in June is 10.2 C. [This object is a pull tab]
Slide 133 / 162 Graphing the Tangent Function Graph by creating a table of values of key points. One option is to use the set of values for x that are multiples of, and the corresponding values of y or tan x.
Slide 134 / 162 Notice that the tangent function is not defined for values of x where cos x = 0, or starting at and every units in either direction. This is shown on the graph by the vertical lines, or asymptotes at these x values. The period of the function is units, because there is one complete cycle from to. As x approaches or any odd multiple of from the left, y increases and approaches infinity. As x approaches from the right, y decreases and approaches negative infinity.
Slide 135 / 162 Example: Sketch the graph of y = tan (x + ) + 2 Asymptotes will be at 0,, 2, etc. The midline will be at y = 2.
Slide 136 / 162 53 Which graph represents y = -tan x? A B C D
Slide 136 (Answer) / 162 53 Which graph represents y = -tan x? A B Answer D C D [This object is a pull tab]
Slide 137 / 162 Trigonometric Identities Return to Table of Contents
Slide 138 / 162 Key Ideas An identity is a mathematical equation that is true for all defined values of the variable. A trigonometric identity is an identity that contains one or more trig ratios. By contrast, a conditional equation is one that is only true for a limited set of values. By learning trig identities, we will be able to replace complicated expressions with simpler ones to solve and verify more difficult equations and identities.
Slide 139 / 162 Drag each equation into the correct box: 3x + 4 = 3x + 4 3x + 4 = 9 5x - 7y = -(7y - 5x) 2x 5 =x 3 sin # + cos # = 1 tan θ cot θ =1 2(x-1) = 2x - 2 (x + 3) 2 = x 2 + 9 sin 4x = 4sin x Identities Conditional Equations
Slide 140 / 162 Reciprocal Identities Basic Trig Identities csc # = 1 sin # sin # = 1 csc # sec # = 1 cos # cos # = 1 sec # cot # = 1 tan # tan # = 1 cot # Tangent Identity tan # = sin # cos # Cotangent Identity cot # = cos # sin #
Slide 141 / 162 By using the basic identities we can change an expression into an equivalent expression. Also, all of the rules of addition, subtraction, multiplication and division that we learned to solve equations and manipulate expressions can be applied to trig expressions and equations.
Slide 142 / 162 Algebraic example Trig example (x - y)(x + y) = x 2 - y 2 (1 - cos #)(1 + cos #) = 1 - cos 2 #
Slide 143 / 162 Pythagorean Identities Recall the unit circle, x 2 + y 2 = 1. (0,1) 1 (cos #,sin #) For any point (x, y) on the circle, its coordinates are (cos #, sin #). Therefore, (-1,0) (1,0) (cos #) 2 + (sin #) 2 = 1 2, which is usually written as (0,-1) cos 2 θ + sin 2 θ = 1
Slide 144 / 162 Pythagorean Identities How do we transform the first identity, which is derived from the unit circle, to the other two?
Slide 145 / 162 Alternative Forms of Identities Since we know that 3 + 5 = 8, we also know that 8-5 = 3 and 8-3 = 5. In elementary school we call these equivalent equations "fact families". Similarly, if cos 2 θ + sin 2 θ = 1, it follows that 1 - cos 2 θ = sin 2 θ and 1 - sin 2 θ = cos 2 θ.
Slide 146 / 162 More Alternative Forms Another fact family tells that since follows that 4 5 = 20. 20 5 = 4, it 1 Since sec θ = cos θ, then sec θ cos θ = 1 (multiply both sides of the first equation by cos #).
Slide 147 / 162 Simplifying Trig Expressions Example 1: Simplify csc θ cos θ tan θ. Rewrite each trig ratio in terms of cos and sin: 1 sin # sin θ cos θ cos # = 1 (When multiplying fractions, it is often easier to reduce or cancel before you multiply.) Example 2: Simplify csc 2 θ(1 - cos 2 θ). 1 sin 2 θ (sin2 θ) = 1
Slide 148 / 162 Verifying an Identity Transform one side of the identity to be the same as the other side Example 1: Verify sin # cot # = cos # sin # cos # = cos # sin # Example 2: Verify cos θ csc θ tan θ = 1 cos # 1 sin # = 1 sin # cos #
Simplify: Slide 149 / 162
Slide 149 (Answer) / 162 Simplify: Answer [This object is a pull tab]
Simplify: Slide 150 / 162
Slide 150 (Answer) / 162 Simplify: Answer [This object is a pull tab]
Simplify: Slide 151 / 162
Slide 151 (Answer) / 162 Simplify: Answer [This object is a pull tab]
Slide 152 / 162
Slide 152 (Answer) / 162
Simplify: Slide 153 / 162
Slide 153 (Answer) / 162 Simplify: Answer [This object is a pull tab]
Verify: Slide 154 / 162
Slide 154 (Answer) / 162 Verify: One possible solution: Answer [This object is a pull tab]
Slide 155 / 162
Slide 155 (Answer) / 162
Verify: Slide 156 / 162
Slide 156 (Answer) / 162 Verify: Answer [This object is a pull tab]
Slide 157 / 162
Slide 157 (Answer) / 162
Slide 158 / 162
Slide 158 (Answer) / 162
Slide 159 / 162 54 Which equation is NOT an identity? A sin 2 x= 1 - cos 2 x B 2 cot x = 2cos x sin x C tan 2 x = sec 2 x - 1 D sin 2 x = cos 2 x - 1
Slide 160 / 162 55 The following expression can be simplified to which choice? A B C D
Slide 160 (Answer) / 162 55 The following expression can be simplified to which choice? A B C D Answer B [This object is a pull tab]
Slide 161 / 162 56 The following expression can be simplified to which choice? A B C D
Slide 161 (Answer) / 162 56 The following expression can be simplified to which choice? A B C D Answer B [This object is a pull tab]
Slide 162 / 162 57 The following expression can be simplified to which choice? A B C D
Slide 162 (Answer) / 162 57 The following expression can be simplified to which choice? A B C D Answer D [This object is a pull tab]